Vector equations of lines and planes (HL only)

IB Mathematics: Analysis and Approaches HL· Unit 3: Geometry & Trigonometry, Topic 10· 25 min read

1. Vector Equation of a Line in 3D★★☆☆☆⏱ 5 min

📘 Definition

Vector equation of a line

$r = a + t d$

General form for a line passing through a point with position vector $a$ , with direction vector $d$ , where $t \in R$ is a scalar parameter that generates all points on the line.

Example:

A line through $(1, 2, 3)$ in direction $(2, - 1, 0)$ is $r = 123 + t 2 - 1 0$

Two other common forms are also used: parametric form (splitting into separate $x, y, z$ components) and Cartesian form (eliminating the parameter $t$ to write a single equality of fractions).

📐 Worked Example

Find the Cartesian equation of the line with vector equation $r = 2 - 1 4 + t 13 - 2$

1
Write the line in parametric form by splitting into components:
$x = 2 + t,\quad y = -1 + 3t,\quad z = 4 - 2t$
2
Isolate $t$ from each equation:
$t = x - 2,\quad t = \frac{y+1}{3},\quad t = \frac{z-4}{-2}$
3
Equate all expressions for $t$ to get the final Cartesian form:
$\frac{x - 2}{1} = \frac{y + 1}{3} = \frac{z - 4}{-2}$

2. Vector Equations of a Plane★★★☆☆HL only⏱ 8 min

📘 Definition

Normal form of a plane

$r \cdot n = d$

A plane with normal vector $n$ (perpendicular to the plane) has this form, where $\frac{d}{∣ n ∣}$ is the perpendicular distance from the origin to the plane. All points with position vector $r$ on the plane satisfy this equation.

Three forms of the plane equation are common in IB exams: normal vector form, Cartesian form (expanded from normal form), and parametric form (using two direction vectors in the plane). If you are given three points on the plane, find two direction vectors by subtracting position vectors, then use the cross product to find the normal vector.

📐 Worked Example

Find the Cartesian equation of the plane passing through points $A (1, 0, 1)$ , $B (2, 1, 3)$ and $C (1, 2, 3)$

1
Find two direction vectors lying on the plane by subtracting point A from B and C:
$\vec{AB} = \begin{pmatrix}1\\1\\2\end{pmatrix}, \quad \vec{AC} = \begin{pmatrix}0\\2\\2\end{pmatrix}$
2
Calculate the normal vector as the cross product of the two direction vectors:
$\mathbf{n} = \vec{AB} \times \vec{AC} = -2\mathbf{i} - 2\mathbf{j} + 2\mathbf{k} = \begin{pmatrix}-2\\-2\\2\end{pmatrix}$
3
Find $d$ by substituting the position vector of point A into normal form:
$d = \mathbf{a} \cdot \mathbf{n} = (1)(-2) + 0(-2) + (1)(2) = 0$
4
Expand and simplify to get Cartesian form:
$-2x -2y + 2z = 0 \implies x + y - z = 0$

3. Intersections of Lines and Planes★★★★☆⏱ 8 min

There are three possible outcomes when finding the intersection of a line and a plane: the line intersects the plane at exactly one point, the line is parallel and never intersects, or the entire line lies on the plane. For two planes, they either intersect along a line or are parallel (distinct or coincident).

📐 Worked Example

Find the intersection point of the line $r = 121 + t 2 - 1 3$ and the plane $2 x + 3 y - z = 7$

1
Write the parametric equations for the line:
$x = 1 + 2t,\quad y = 2 - t,\quad z = 1 + 3t$
2
Substitute the parametric expressions into the plane equation:
$2(1+2t) + 3(2-t) - (1+3t) = 7$
3
Expand and solve for $t$ :
$2 + 4t + 6 - 3t - 1 - 3t = 7 \implies 7 - 2t = 7 \implies t = 0$
4
Substitute $t = 0$ back to find the intersection point:
$(1, 2, 1)$

4. Angles Between Lines and Planes★★★★☆⏱ 4 min

All angles between 3D vector objects are calculated using the dot product, with different formulas depending on what you are finding the angle between. Angles are always between $0^{\circ}$ and $9 0^{\circ}$ in this topic.

📘 Definition

Angle between two planes

The acute angle between two planes is equal to the acute angle between their normal vectors, calculated with the formula:

Example:

$cos θ = \frac{∣ n _{1} \cdot n _{2} ∣}{∣ n _{1} ∣∣ n _{2} ∣}$

📐 Worked Example

Find the angle between the planes $x + y + z = 1$ and $2 x - y + 3 z = 2$ , to the nearest degree.

1
Extract the normal vectors from the Cartesian equations:
$\mathbf{n_1} = \begin{pmatrix}1\\1\\1\end{pmatrix}, \quad \mathbf{n_2} = \begin{pmatrix}2\\-1\\3\end{pmatrix}$
2
Calculate the dot product and magnitudes of the normals:
$\mathbf{n_1} \cdot \mathbf{n_2} = 4, \quad |\mathbf{n_1}| = \sqrt{3}, \quad |\mathbf{n_2}| = \sqrt{14}$
3
Substitute into the formula to find $θ$ :
$\cos\theta = \frac{|4|}{\sqrt{3}\sqrt{14}} \approx 0.617 \implies \theta \approx 52^\circ$

5. Common Pitfalls

Wrong move:

Confusing position and direction vectors when writing a line equation

Why:

Swapping these gives an incorrect line through the origin instead of the given point

Correct move:

Always keep the position vector of your known point as the constant term, and multiply the direction vector by the parameter.

Wrong move:

Forgetting the absolute value in angle formulas

Why:

A negative dot product will give you the obtuse supplementary angle, instead of the required acute angle between objects

Correct move:

Always take the absolute value of the dot product when calculating angles between lines and planes.

Wrong move:

Assuming planes with the same normal are the same plane

Why:

Planes with the same normal are parallel but can be distinct if the constant term $d$ is different

Correct move:

Check the value of $d$ from $r \cdot n = d$ to confirm if parallel planes are coincident or distinct.

Wrong move:

Calculating direction of intersection line with dot product of normals

Why:

The direction of the intersection line must be perpendicular to both normals, which requires cross product not dot product

Correct move:

The direction vector of the line of intersection of two planes is always the cross product of the two normals.

Wrong move:

Swapping normal component order when converting to Cartesian plane equation

Why:

This mixes up the coefficients of $x, y, z$ and gives an incorrect plane

Correct move:

Always match the first component of the normal to $x$ , the second to $y$ , and the third to $z$ .

6. Quick Reference Cheatsheet

Object	Formula	Notes
Line in 3D	$r = a + t d$	Cartesian: $\frac{x - a _{x}}{d _{x}} = \frac{y - a _{y}}{d _{y}} = \frac{z - a _{z}}{d _{z}}$
Plane (normal form)	$r \cdot n = d$	Cartesian: $n_{x} x + n_{y} y + n_{z} z = d$
Plane (parametric)	$r = a + s d_{1} + t d_{2}$	Convert via $n = d_{1} \times d_{2}$
Angle between two lines	$cos θ = \frac{∣ d _{1} \cdot d _{2} ∣}{∣ d _{1} ∣∣ d _{2} ∣}$	$d_{1}, d_{2}$ = direction vectors
Angle between two planes	$cos θ = \frac{∣ n _{1} \cdot n _{2} ∣}{∣ n _{1} ∣∣ n _{2} ∣}$	$n_{1}, n_{2}$ = normal vectors
Angle between line and plane	$sin θ = \frac{∣ d \cdot n ∣}{∣ d ∣∣ n ∣}$	$d$ = line direction, $n$ = plane normal

7. Frequently Asked

Do I need to memorize all forms of the plane equation?

Yes, exam questions regularly require switching between vector, parametric, and Cartesian forms depending on the given information.

How do I know if a line is parallel to a plane?

A line is parallel to a plane if its direction vector is perpendicular to the plane's normal vector, meaning their dot product equals zero.

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2025 · 1
Plane intersection, 8 marks
2024 · 2
Angle between planes, 6 marks
2023 · 1
Line-plane intersection, 5 marks

Going deeper

syllabusIB Official AA HL SyllabusFull topic specification

What's Next

Mastering vector equations of lines and planes is the foundation for all advanced 3D vector topics tested in IB AA HL. The next core skill is calculating distances from points to lines and planes, which relies entirely on your ability to write correct vector equations for these objects. This topic also prepares you for more extended problems involving intersections of multiple planes, which regularly appear as high-mark questions in Paper 1 and Paper 2 of the exam.